Sets
As mentioned in the introduction, a fundamental concept in all a branch of mathematics is that of set. Here is a definition “A set is any well-defined list, collection or class of objects”.
The objects in sets, as we shall see from examples, can be anything: But for clarity, we now list ten particular examples of sets:
Example 1.1 The numbers 0, 2, 4, 6, 8
Example 1.2 The solutions of the equation x²+ 2x+1 = 0
Example 1.3 The vowels of the alphabet: a, e, i, o, u
Example 1.4 The people living on earth
Example 1.5 The students Tom, Dick and Harry
Example1.6 The students who are absent from school
Example 1.7 The countries England, France and Denmark
Example 1.8 The capital cities of Nigeria
Example 1.9 The number 1, 3, 7, and 10
Example 1.10 The Rivers in Nigeria
Note that the sets in the odd numbered examples are defined, that is, presented, by actually listing its members; and the sets in the even numbered examples are defined by stating properties that is, rules, which decide whether or not a particular object is a member of the set
13. AXIOMATIC DEVELOPMENT OF SET THEORY
In an axiomatic development of a branch of mathematics, one begins with:
1. Undefined terms
2. Undefined relations
3. Axioms relating the undefined terms and undefined relations. Then, one develops theorems based upon the axioms and definitions Example 14:1:
In an axiomatic development of Plane Euclidean geometry
1. “Points” and “lines” are undefined terms
2. “Points on a line” or, equivalent, “line contain a point” is an undefined relation
3. Two of the axioms are:
Axiom 1: Two different points are on one and only one line
Axiom 2: Two different lines cannot contain more than one point in common
In an axiomatic development of set theory:
1. “Element’ and “set” are undefined terms
2. “Element belongs to a set” is undefined relation
3. Two of the axioms are
Axiom of Extension: Two sets A and B are equal if and only if every element in A belongs to B and every element in B belongs to A.
Axiom of Specification: Let P(x) be any statement and let A be any set. Then there exists a set:
B= {aa ∈ A, P (a) is true}
Here, P(x) is a sentence in one variable for which P(a) is true or false for any a∈A. for example P(x) could be the sentence “x² = 4” or “x is a member of the United Nations”